Two-Phase Network Data Envelopment Analysis: An Example of Bank Performance Assessment Two-Phase Network Data Envelopment Analysis: An Example of Bank Performance Assessment

Data envelopment analysis (DEA) models assess decision-making units (DMUs), which directly convert multiple inputs into multiple outputs. Network DEA models have been studied extensively. However, the performance indices that link the two stages are assumed to be fixed or non-discretionary; their values are not adjustable. These models only assumed that the reductions on the inputs and additions on the outputs would improve the overall efficiency. But in the real world, the link is always adjustable. “ Free links ” means that the intermediate items are adjustable or discretionary, and each DMU can be increased or decreased from the observed one. The current chapter introduces a two-phase procedure with free links to assess system performance, Phase-I is a proposed slack-based measurement (SBM) model to partition the links into two sets: as-input and as-output. Phase-II is a modified SBM model to determine the slack of each input, as-input link, output and as-output link. This proposed model counts the slacks associated with the intermediate items in the efficiency scores and determines the entire system performance by the directional distance function. It is validated using network procedure and assesses the performance of supply chain management system.


Introduction
The data envelopment analysis (DEA) models assess a set of homogeneous decision-making units (DMUs) that convert inputs into outputs. Fewer input values and more output values are desired and DMUs may be classified as being either efficient or inefficient. Tone and Tsutsui [1,2] introduce network and dynamic DEA and categorize links into two types-"fixed links" However, these approaches do not count the slacks associated with the intermediate items in the efficiency scores. Consequently, the efficiency scores are greater than the actual efficiency. In addition, there is no DMU with an efficiency score equal to 1 because the properties of intermediate performance evaluation items would lead to conflicts. For instance, in the twostage process problem, Stage-2 may have to reduce inputs (links) to achieve an efficient status. However, doing so would lead to a reduction in outputs in Stage-1, thereby reducing the efficiency of Stage-1. In other words, there are still two efficiency frontiers for the two subprocesses. One may desire a single frontier for the entire production system. "Link" cannot be adjusted freely in a radial model which adjusts the inputs and outputs by the efficiency scores in a two-stage process. For this model, the entire system efficiency cannot be improved by adjusting links, see Kao and Hwang [10] and Lewis and Sexton [9]. "Link" that applies in a non-radial model has been discussed in recent years. Tone and Tsutsui [1] introduce a network DEA and categorize links into two types-"fixed links" and "free links." "Free links" means the intermediate items are adjustable or discretionary; each DMU can be increased or decreased from the observed one and is free to assign each individual link to one of the three characteristics: as-input, as-output, or non-discretionary so that the entire system efficiency could be maximized. "Fixed links" means the intermediate products are beyond the control of DMUs. In the radial model, "links" cannot be adjusted freely, which adjust the inputs and outputs by the efficiency scores in a two-stage process. Tone and Tsutsui [2] introduce the dynamic slack-based measure (DSBM) model and the incorporation of slacks with free and fixed links into the efficiency score. They categorize the links into four types: desirable, undesirable, discretionary (free), and non-discretionary (fixed). The article incorporates the slacks of free links into the efficiency score in two ways: an ex-post approach (adjusted score) and incorporation through 0-1 MIP. The ex-post approach includes a two-phase procedure. Tone and Tsutsui [1] introduce the links are discretionary regarding their status, as-input or as-output. Liu and Liu [16,17] adopt VGM and GBM models to assess the performance of supply chain management.
Chambers et al. [18] introduced the directional distance function (DDF) based on the Luenberger benefit function to obtain the technical efficiency by increasing the outputs and reducing the inputs simultaneously. Later, Chambers et al. [19] introduced the DDF of DEA to measure the technical efficiency. This chapter develops a model for an improved efficiency measure through directional distance formulation of data envelopment analysis.
The contribution and innovative progress for this chapter are (1) creating a new SBM model and converting multi-efficiency frontiers for the separation processes to an aggregation efficiency frontier for the entire production system and (2) adopting free links application and introducing DDF with a virtual gap diagram to assess the performance of the entire system. The rest of this chapter is organized as follows. The proposed two-phase two-stage performance evaluation models and DDF are presented in Section 2. Because the uniqueness of the optimal solution is important, we report an experiment on this subject using a real-world bank performance assessment in Section 3. We conclude this chapter in the last section.
2. The proposed two-phase two-stage performance evaluation J denotes the set of homogeneous decision-making units of a network process that are evaluated by a set of inputs, I, a set of free links, D free , and a set of outputs R. DMU o represents the DMU under evaluation. To maximize the system efficiency score of DMU o , each link in set D free is "free" to be assigned to one of the subsets -D À o , D free o , and D þ o if it is as-input, free link, and as-output, respectively. Figure 1 depicts the two-phase procedure to evaluate the performance of DMUs using the two-stage and network processes. This two-phase procedure contains two slack-based DEA linear programming models. Phase-I sets all links in set D free to discretionary and the objective is to determine the maximum slack values on each input and output so that the weights of each DMU in each stage can be assigned. The set D free is partitioned into two The output of Phase-I will indicate that several links in set D free o should be assigned to sets D À o and D þ o . The target of Phase-II is thus to determine the maximum reduction value on each input and link in sets I and D À o and the addition value on each output and link in sets R and D þ o such that the weights of each DMU in the system can be assigned. The target of each input, free link, and output on the frontier is identified.  DMU o . The first task is to assign all elements in the set to either D þ o or D À o . The second task is to place the slacks of link d in sets and add them to its aggregate performance score. Phase-II of our solving procedure addresses the first task. [M1] (1) s:t: For the two-phase procedure which is depicted in Figure 1 o . The decision variables π 1 j and π 2 j denote the weights in Stage-1 and Stage-2, respectively, of DMU j in evaluating DMU o . [M2] s:t:   [21]) that measures the efficiency of converting the sets of input and as-input indices I ∪D À o into the sets of output and as-output indices R ∪D þ o . The frontiers of π 1 j and π 2 j are converted into an entire system frontier π j in this phase. [M3] s:t: The dual form of [M3] is expressed as [M4]. The decision variables of [M4] possess properties and , representing the weight assigned to the ith input and the rth output, respectively. The terms w þ d and w À d represent the weight assigned to link d in sets D þ o and D À o , respectively. [M4] Inequality (4.2) may be revised such that P r ∈ R u r y rj þ , and the constraint ensures that the maximum performance value of each DMU j is not greater than 1.

Proposed directional distance function approach
The directional distance function (DDF) measures the distance from a certain operation point (e.g., DMU o ) to the efficient frontier of the technology along the positive semi-ray defined by vector g. Given a directional vector g ¼ Àg À The objective function (4.1) can be modeled by using the DDF. We denote virtual input by r ∈ R u r y ro ) which are identified by specifying a directional vector g. The objective function (4.1) can be converted to (4.9) which is to minimize the virtual input and maximize the virtual output to reach the efficient frontier.
The graph technology can be represented by The optimal solution of virtual gap Δ * o expresses as DDF: This chapter defines a virtual gap diagram; the summation of input and as-input is the x-axis ) and the summation of output and as-output is the y-axis It is obvious that the minimum virtual gap " " is equivalent to the maximum efficiency score of the entire network. It ensures that the nearest improvement target is found. Figure 2 depicts the virtual gap diagram; x-axis denotes the virtual input and y-axis denotes the virtual output.

Overall stage efficiencies
Similar to the SBM non-oriented models of Tone and Tsutsui [20], the solutions of Phase-II provide a reference set of DMUs for DMU o . The target for the performance items in sets I,D þ o , Figure 2. Virtual gap diagram. D À o , and R can be obtained using [E1]. The measured performance value E * o is the best practice for DMU o in the overall two-stage process which is expressed as These points are the projection of DMU o on the frontier.
The results of Phase-II, s À * i , s À * zd , s þ * zd and s þ * r , are used to compute the efficiencies of Stage-1, E * 1 , and Stage-2; E * 2 is shown in the following two Eqs. (E3 and E4). For the efficiencies of Stage-1, the numerator is the summation of inputs (x io , i∈ I) and as-input (z do , d∈ D À o ). The denominator is the as-output items (z do , d∈ D þ o ). Likewise, for the efficiencies of Stage-2, the numerator is the as-input item (z do , d∈ D À o ). The denominator is the summation of the as-output (z do , d∈ D þ o ) and outputs (y ro , r∈ R).
If set is empty, the denominator is equal to 1.
If set is empty, the numerator is equal to 1.

From Eqs. [E3] and [E4]
, we obtain the performance scores of Stage-1 and Stage-2, respectively, which identify the performance of each stage.

To extend two-stage to network process
Liu and Liu [16] extend the two-stage to network process. The network contains a set of subprocesses (nodes), H. The nodes are assigned ordinal numbers 1, 2, 3,…, n. Let A denote the set of network links. There are n homogeneous DMUs in set J, named DMU 1 , DMU 2 ,…, and DMU n , which are randomly processed by the sub-processes in set H. The network structure is depicted in Figure 3.

Inputs and outputs
At each sub-process h, there is a set of input measures I h that flow into the network and a set of output measures R h that flow out of the network. For DMU j in set J, let x h ij ∈ ℜ I h þ and y h rj ∈ ℜ R h þ denote the volumes of the ith input measure and the rth output measure at the sub-process h, respectively. Let s hÀ i and s hþ r be the slack of the ith input and the rth output at sub-process h, respectively.

Links
Each sub-process may have links to other sub-processes. Let (h, k) denote the link between sub-

Illustrative examples
This study adopts a dataset covering 24 non-life insurance companies in Taiwan from Kao and Hwang [10] to illustrate the proposed two-phase procedure. Table 1 summarizes the performance datasheet of 24 non-life insurance companies in Taiwan. The inputs of the system: Operation expenses (x 1 ): salaries of the employees and various types of costs incurred in daily operation and.
Insurance expenses (x 2 ): expenses paid to agencies, brokers, and solicitors and other expenses associated with marketing the service of insurance.
The links of the system: Direct written premiums (z 1 ): premiums received from insured clients.
Reinsurance premiums (z 2 ): premiums received from ceding companies.  The outputs of the system: Under-writing profit (y 1 ): profit earned from the insurance business.
Investment profit (y 2 ): profit earned from the investment portfolio.
3.1. Phase-I Table 2 summarizes the results of Phase-I and Phase-II. In the Phase-I column, for example, when DMU 1 is being evaluated, DMU o = DMU 1 and the optimal solution of [M1] is s þ * z1 ¼ 0, s þ * z2 ¼ 549, 067, s À * z1 ¼ 877, 494, and s À *  =0. This calculation indicates that the natural link, d = 1, acts as an "asinput" item and may have a better solution. Therefore, 1(3,174,850) is recorded under the column of Phase-I. DMU 7 and DMU 18 have solution processes that are similar to DMU 4 .

Phase-II
Because each link may be "as-input" or "as-output", the two links may have four possible combinations of D þ o and D À o . Table 4 shows the four categories A, B, C, and D and their link settings. As indicated in the first column of Table 3, 15, 3, 3, and 3 DMUs belong to Categories A, B, C, and D, respectively. For instance, DMU 4 in Category A treats the first links as "asinput" (slack = 1,072,937) and is an undesirable output with respect to Stage-1 and a desirable input with respect to Stage-2. Meanwhile, the second set of links is "as-output" (slack = 135,818) and represents a desirable output with respect to Stage-1 but an undesirable input with respect to Stage-2.
Proceeding to Phase-II, which employs [M5], the optimal solutions for the evaluated DMU are listed in Table 4 The virtual weight is expressed as v * i x ij = P Virtual Input, w À * d z dj = P Virtual Input w þ * d z dj = P Virtual Output and u * r y rj =  input or "as-input" link in the overall virtual input weight and the percentage of each output or "asoutput" link in the overall virtual output weight. As shown at  Stage-2. An obvious means of improving overall efficiency is to focus on Stage-2 output item u 2 y * 2 ; it is 85% of output and as-output items.

Discussion and conclusions
The objective of efficiency assessment is to identify weaknesses such that the appropriate steps to improve the entire system's performance. This chapter introduces a two-phase procedure to evaluate the two-stage and network models with "free" links. This new model adopts SBM   and considers not only the input and output slacks in the objective function but also the slacks of links. The resultant DEA scores provide completely information on how to project inefficient DMUs onto the DEA frontier for specific two-stage processes. Instead of the two conflicting roles that each link plays in existing models, each link plays a single role in the proposed two-phase process system in that it is either desirable or undesirable. The SBM model in this chapter counts the slacks associated with links in the efficiency scores, overcoming the hurdle. The bank case study takes the example on adjustment in the slacks and defines the best practice performance that the DMU under evaluation will need to attain to achieve the best efficiency. To achieve the best-practice efficiency, each DMU determines a set of weights for input, output, and link, where the links are designated as either "as-input" or "as-output". Input and as-input measures reduce slacks, while output and as-output measures increase slacks to reach their targets on the production frontier. This study only introduces a two-stage procedure to assess the entire system. It also can be extended to more complex network processes, applied in series multistage, share resource (Chen et al. [21] and Liang et al. [22]), dynamic network DEA (Tone & Tsutsui [2] and Kao [13]), assurance region (Thompson et al. [23]), cone ratio model (Charnes et al. [24]), and virtual weight analysis models (Sarrico & Dyson [25]) in future research.